We study the low-temperature transport properties of clean one-dimensional spin-1 2 chains coupled to phonons. Due to the presence of approximate conservation laws, the heat current decays very slowly giving rise to an exponentially large heat conductivity, κ∼ e T*/T. As a result of an interplay of umklapp scattering and spinon-phonon coupling, the characteristic energy scale T* turns out to be of order Θ D/2, where Θ D is the Debye energy, rather than the magnetic exchange interaction J—in agreement with recent measurements in SrCuO compounds. A large magnetic field h strongly affects the heat transport by two distinct mechanisms. First, it induces a linear spinon-phonon coupling, which alters the nature of the T→ 0 fixed point: the elementary excitations of the system are composite spinon-phonon objects. Second, the change of the magnetization and the corresponding change of the wave vector of the spinons strongly affects the way in which various umklapp processes can relax the heat current, leading to a characteristic fractal-like spiky behavior of κ (T, h) as a function of h.